410 14. EQUATIONS AND ALGORITHMScould provide satisfactory proofs. If this was the opinion of a lead-
ing mathematician of the generation just preceding the birth of
the axiomatic method, then it is rather obvious that early Greek
mathematics cannot have been very different from the Heronic Dio-
phantine type.1.1. Diophantine equations. An equation containing two or more unknowns for
which only rational (or more often, integer) solutions are sought is nowadays called
a Diophantine equation. Diophantus wrote a treatise commonly known under the
somewhat misleading name Arithmetica. As mentioned previously, the six books of
this treatise that have been known for some centuries may now be supplemented
by parts of four other books, discovered in 1968, but that is not certain. The title
itself is of some interest. Its suffix -tica has come into English from Greek in a large
number of words such as logistics, mathematics, and gymnastics. It has a sense of
how-to, that is, the techniques involved in using numbers (arithmoi) or reasoning
(logoi) or learning (mathemata) or physical training (gymnasts).^1 The plural -s
on these English words, even though they are now regarded as singular, reflects
the fact that these words were originally intended to be plural—the neuter plural
form of the corresponding adjectives arithmetikos (adept with number), logistikos
(skilled in calculating), mathematikos (disposed to learn), and gymnastikos (skilled
in bodily exercise), but which evolved into a feminine singular form. The Greek
title of the work is Diophantou Alexdndreds ArithmMikon, meaning [The Books] of
Arithmetics of Diophantus of Alexandria.
1 .2. General characteristics of the Arithmetica. In contrast to other ancient
works containing problems that lead to algebra, the problems that require algebraic
techniques in the Arithmetica all involve purely numerical relations. They are
not problems about things that have been counted or measured. They are about
counting itself. The work begins with a note to one Dionysius, whom the author
characterizes as "eager to learn" how to solve problems in arithmetic.^2 In a number
of ways Diophantus seems to be doing something that resembles the algebra taught
nowadays. In particular, he has a symbol for an unknown or abstract number that
is to be found in a problem, and he appears to know what an equation is, although
he doesn't exactly use the word equation.
ï
Diophantus began by introducing a symbol for a constant unit M, from monas
(ìïíÜò), along with a symbol for an unknown number ò, conjectured to be an
abbreviation of the first two letters of the Greek word for number: arithmos
(áñéèìüò). For the square of an unknown he used Av, the first two letters of
dynamis (Áííáìé,ò), meaning power. For its cube he used Kv, the first two letters
of kybos (Êýâïò), meaning cube. He then combined these letters to get fourth
(ÄõÄ), fifth (ÄÊõ), and sixth (KVK) powers. For the reciprocals of these powers
of the unknown he invented names by adjoining the suffix -ton (-ôïí) to the names
of the corresponding powers. These various powers of the unknown were called
eida (åßäá), meaning species. Diophantus' system for writing down the equivalent
of a polynomial in the unknown consisted of writing down these symbols in order
(^1) From the root gymnos, meaning naked.
(^2) One of the reasons that Tannery assigned Dionysius to the third century was that this date
made it easy to imagine that Dionysius was the man appointed Bishop of Alexandria in 247—not
that Dionysius (Dennis) was exactly a rare name in those days.